UNITARY REPRESENTATIONS and COMPLEX ANALYSIS David A
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Physics.Hist-Ph] 15 May 2018 Meitl,Fo Hsadsadr Nepeainlprinciples
Why Be Regular?, Part I Benjamin Feintzeig Department of Philosophy University of Washington JB (Le)Manchak, Sarita Rosenstock, James Owen Weatherall Department of Logic and Philosophy of Science University of California, Irvine Abstract We provide a novel perspective on “regularity” as a property of representations of the Weyl algebra. We first critique a proposal by Halvorson [2004, “Complementarity of representa- tions in quantum mechanics”, Studies in History and Philosophy of Modern Physics 35(1), pp. 45–56], who argues that the non-regular “position” and “momentum” representations of the Weyl algebra demonstrate that a quantum mechanical particle can have definite values for position or momentum, contrary to a widespread view. We show that there are obstacles to such an intepretation of non-regular representations. In Part II, we propose a justification for focusing on regular representations, pace Halvorson, by drawing on algebraic methods. 1. Introduction It is standard dogma that, according to quantum mechanics, a particle does not, and indeed cannot, have a precise value for its position or for its momentum. The reason is that in the standard Hilbert space representation for a free particle—the so-called Schr¨odinger Representation of the Weyl form of the canonical commutation relations (CCRs)—there arXiv:1805.05568v1 [physics.hist-ph] 15 May 2018 are no eigenstates for the position and momentum magnitudes, P and Q; the claim follows immediately, from this and standard interpretational principles.1 Email addresses: [email protected] (Benjamin Feintzeig), [email protected] (JB (Le)Manchak), [email protected] (Sarita Rosenstock), [email protected] (James Owen Weatherall) 1Namely, the Eigenstate–Eigenvalue link, according to which a system has an exact value of a given property if and only if its state is an eigenstate of the operator associated with that property. -
Lie Algebras and Representation Theory Andreasˇcap
Lie Algebras and Representation Theory Fall Term 2016/17 Andreas Capˇ Institut fur¨ Mathematik, Universitat¨ Wien, Nordbergstr. 15, 1090 Wien E-mail address: [email protected] Contents Preface v Chapter 1. Background 1 Group actions and group representations 1 Passing to the Lie algebra 5 A primer on the Lie group { Lie algebra correspondence 8 Chapter 2. General theory of Lie algebras 13 Basic classes of Lie algebras 13 Representations and the Killing Form 21 Some basic results on semisimple Lie algebras 29 Chapter 3. Structure theory of complex semisimple Lie algebras 35 Cartan subalgebras 35 The root system of a complex semisimple Lie algebra 40 The classification of root systems and complex simple Lie algebras 54 Chapter 4. Representation theory of complex semisimple Lie algebras 59 The theorem of the highest weight 59 Some multilinear algebra 63 Existence of irreducible representations 67 The universal enveloping algebra and Verma modules 72 Chapter 5. Tools for dealing with finite dimensional representations 79 Decomposing representations 79 Formulae for multiplicities, characters, and dimensions 83 Young symmetrizers and Weyl's construction 88 Bibliography 93 Index 95 iii Preface The aim of this course is to develop the basic general theory of Lie algebras to give a first insight into the basics of the structure theory and representation theory of semisimple Lie algebras. A problem one meets right in the beginning of such a course is to motivate the notion of a Lie algebra and to indicate the importance of representation theory. The simplest possible approach would be to require that students have the necessary background from differential geometry, present the correspondence between Lie groups and Lie algebras, and then move to the study of Lie algebras, which are easier to understand than the Lie groups themselves. -
Structure Theory of Finite Conformal Algebras Alessandro D'andrea JUN
Structure theory of finite conformal algebras by Alessandro D'Andrea Laurea in Matematica, Universith degli Studi di Pisa (1994) Diploma, Scuola Normale Superiore (1994) Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1998 @Alessandro D'Andrea, 1998. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part. A uthor .. ................................ Department of Mathematics May 6, 1998 Certified by ............................ Victor G. Kac Professor of Mathematics rc7rc~ ~ Thesis Supervisor Accepted by. Richard B. Melrose ,V,ASSACHUSETT S: i i. Chairman, Department Committee OF TECHNOLCaY JUN 0198 LIBRARIES Structure theory of finite conformal algebras by Alessandro D'Andrea Submitted to the Department ,of Mathematics on May 6, 1998, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis I gave a classification of simple and semi-simple conformal algebras of finite rank, and studied their representation theory, trying to prove or disprove the analogue of the classical Lie algebra representation theory results. I re-expressed the operator product expansion (OPE) of two formal distributions by means of a generating series which I call "A-bracket" and studied the properties of the resulting algebraic structure. The above classification describes finite systems of pairwise local fields closed under the OPE. Thesis Supervisor: Victor G. Kac Title: Professor of Mathematics Acknowledgments The few people I would like to thank are those who delayed my thesis the most. -
Fourier Analysis Using Representation Theory
FOURIER ANALYSIS USING REPRESENTATION THEORY NEEL PATEL Abstract. In this paper, it is our goal to develop the concept of Fourier transforms by using Representation Theory. We begin by laying basic def- initions that will help us understand the definition of a representation of a group. Then, we define representations and provide useful definitions and the- orems. We study representations and key theorems about representations such as Schur's Lemma. In addition, we develop the notions of irreducible repre- senations, *-representations, and equivalence classes of representations. After doing so, we develop the concept of matrix realizations of irreducible represen- ations. This concept will help us come up with a few theorems that lead up to our study of the Fourier transform. We will develop the definition of a Fourier transform and provide a few observations about the Fourier transform. Contents 1. Essential Definitions 1 2. Group Representations 3 3. Tensor Products 7 4. Matrix Realizations of Representations 8 5. Fourier Analysis 11 Acknowledgments 12 References 12 1. Essential Definitions Definition 1.1. An internal law of composition on a set R is a product map P : R × R ! R Definition 1.2. A group G, is a set with an internal law of composition such that: (i) P is associative. i.e. P (x; P (y; z)) = P (P (x; y); z) (ii) 9 an identity; e; 3 if x 2 G; then P (x; e) = P (e; x) = x (iii) 9 inverses 8x 2 G, denoted by x−1, 3 P (x; x−1) = P (x−1; x) = e: Let it be noted that we shorthand P (x; y) as xy. -
Factorization of Unitary Representations of Adele Groups Paul Garrett [email protected]
(February 19, 2005) Factorization of unitary representations of adele groups Paul Garrett [email protected] http://www.math.umn.edu/˜garrett/ The result sketched here is of fundamental importance in the ‘modern’ theory of automorphic forms and L-functions. What it amounts to is proof that representation theory is in principle relevant to study of automorphic forms and L-function. The goal here is to get to a coherent statement of the basic factorization theorem for irreducible unitary representations of reductive linear adele groups, starting from very minimal prerequisites. Thus, the writing is discursive and explanatory. All necessary background definitions are given. There are no proofs. Along the way, many basic concepts of wider importance are illustrated, as well. One disclaimer is necessary: while in principle this factorization result makes it clear that representation theory is relevant to study of automorphic forms and L-functions, in practice there are other things necessary. In effect, one needs to know that the representation theory of reductive linear p-adic groups is tractable, so that conversion of other issues into representation theory is a change for the better. Thus, beyond the material here, one will need to know (at least) the basic properties of spherical and unramified principal series representations of reductive linear groups over local fields. The fact that in some sense there is just one irreducible representation of a reductive linear p-adic group will have to be pursued later. References and historical notes -
A Gentle Introduction to a Beautiful Theorem of Molien
A Gentle Introduction to a Beautiful Theorem of Molien Holger Schellwat [email protected], Orebro¨ universitet, Sweden Universidade Eduardo Mondlane, Mo¸cambique 12 January, 2017 Abstract The purpose of this note is to give an accessible proof of Moliens Theorem in Invariant Theory, in the language of today’s Linear Algebra and Group Theory, in order to prevent this beautiful theorem from being forgotten. Contents 1 Preliminaries 3 2 The Magic Square 6 3 Averaging over the Group 9 4 Eigenvectors and eigenvalues 11 5 Moliens Theorem 13 6 Symbol table 17 7 Lost and found 17 References 17 arXiv:1701.04692v1 [math.GM] 16 Jan 2017 Index 18 1 Introduction We present some memories of a visit to the ring zoo in 2004. This time we met an animal looking like a unicorn, known by the name of invariant theory. It is rare, old, and very beautiful. The purpose of this note is to give an almost self contained introduction to and clarify the proof of the amazing theorem of Molien, as presented in [Slo77]. An introduction into this area, and much more, is contained in [Stu93]. There are many very short proofs of this theorem, for instance in [Sta79], [Hu90], and [Tam91]. Informally, Moliens Theorem is a power series generating function formula for counting the dimensions of subrings of homogeneous polynomials of certain degree which are invariant under the action of a finite group acting on the variables. As an apetizer, we display this stunning formula: 1 1 ΦG(λ) := |G| det(id − λTg) g∈G X We can immediately see elements of linear algebra, representation theory, and enumerative combinatorics in it, all linked together. -
SCHUR-WEYL DUALITY Contents Introduction 1 1. Representation
SCHUR-WEYL DUALITY JAMES STEVENS Contents Introduction 1 1. Representation Theory of Finite Groups 2 1.1. Preliminaries 2 1.2. Group Algebra 4 1.3. Character Theory 5 2. Irreducible Representations of the Symmetric Group 8 2.1. Specht Modules 8 2.2. Dimension Formulas 11 2.3. The RSK-Correspondence 12 3. Schur-Weyl Duality 13 3.1. Representations of Lie Groups and Lie Algebras 13 3.2. Schur-Weyl Duality for GL(V ) 15 3.3. Schur Functors and Algebraic Representations 16 3.4. Other Cases of Schur-Weyl Duality 17 Appendix A. Semisimple Algebras and Double Centralizer Theorem 19 Acknowledgments 20 References 21 Introduction. In this paper, we build up to one of the remarkable results in representation theory called Schur-Weyl Duality. It connects the irreducible rep- resentations of the symmetric group to irreducible algebraic representations of the general linear group of a complex vector space. We do so in three sections: (1) In Section 1, we develop some of the general theory of representations of finite groups. In particular, we have a subsection on character theory. We will see that the simple notion of a character has tremendous consequences that would be very difficult to show otherwise. Also, we introduce the group algebra which will be vital in Section 2. (2) In Section 2, we narrow our focus down to irreducible representations of the symmetric group. We will show that the irreducible representations of Sn up to isomorphism are in bijection with partitions of n via a construc- tion through certain elements of the group algebra. -
23 Infinite Dimensional Unitary Representations
23 Infinite dimensional unitary representations Last lecture, we found the finite dimensional (non-unitary) representations of SL2(R). 23.1 Background about infinite dimensional representa- tions (of a Lie group G) What is an infinite dimensional representation? 1st guess Banach space acted on by G? This is no good for the following reasons: Look at the action of G on the functions on G (by left translation). We could use L2 functions, or L1 or Lp. These are completely different Banach spaces, but they are essentially the same representation. 2nd guess Hilbert space acted on by G? This is sort of okay. The problem is that finite dimensional representations of SL2(R) are NOT Hilbert space representations, so we are throwing away some interesting representations. Solution (Harish-Chandra) Take g to be the Lie algebra of G, and let K be the maximal compact subgroup. If V is an infinite dimensional representation of G, there is no reason why g should act on V . The simplest example fails. Let R act on L2(R) by left translation. Then d d 2 R d the Lie algebra is generated by dx (or i dx ) acting on L ( ), but dx of an L2 function is not in L2 in general. Let V be a Hilbert space. Set Vω to be the K-finite vectors of V , which are the vectors contained in a finite dimensional representation of K. The point is that K is compact, so V splits into a Hilbert space direct sum finite dimensional representations of K, at least if V is a Hilbert space. -
Generalized Supercharacter Theories and Schur Rings for Hopf Algebras
Generalized Supercharacter Theories and Schur Rings for Hopf Algebras by Justin Keller B.S., St. Lawrence University, 2005 M.A., University of Colorado Boulder, 2010 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics 2014 This thesis entitled: Generalized Supercharacter Theories and Schur Rings for Hopf Algebras written by Justin Keller has been approved for the Department of Mathematics Nathaniel Thiem Richard M. Green Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii Keller, Justin (Ph.D., Mathematics) Generalized Supercharacter Theories and Schur Rings for Hopf Algebras Thesis directed by Associate Professor Nathaniel Thiem The character theory for semisimple Hopf algebras with a commutative representation ring has many similarities to the character theory of finite groups. We extend the notion of superchar- acter theory to this context, and define a corresponding algebraic object that generalizes the Schur rings of the group algebra of a finite group. We show the existence of Hopf-algebraic analogues for the most common supercharacter theory constructions, specificially the wedge product and super- character theories arising from the action of a finite group. In regards to the action of the Galois group of the field generated by the entries of the character table, we show the existence of a unique finest supercharacter theory with integer entries, and describe the superclasses for abelian groups and the family GL2(q). -
Arxiv:1106.4415V1 [Math.DG] 22 Jun 2011 R,Rno Udai Form
JORDAN STRUCTURES IN MATHEMATICS AND PHYSICS Radu IORDANESCU˘ 1 Institute of Mathematics of the Romanian Academy P.O.Box 1-764 014700 Bucharest, Romania E-mail: [email protected] FOREWORD The aim of this paper is to offer an overview of the most important applications of Jordan structures inside mathematics and also to physics, up- dated references being included. For a more detailed treatment of this topic see - especially - the recent book Iord˘anescu [364w], where sugestions for further developments are given through many open problems, comments and remarks pointed out throughout the text. Nowadays, mathematics becomes more and more nonassociative (see 1 § below), and my prediction is that in few years nonassociativity will govern mathematics and applied sciences. MSC 2010: 16T25, 17B60, 17C40, 17C50, 17C65, 17C90, 17D92, 35Q51, 35Q53, 44A12, 51A35, 51C05, 53C35, 81T05, 81T30, 92D10. Keywords: Jordan algebra, Jordan triple system, Jordan pair, JB-, ∗ ∗ ∗ arXiv:1106.4415v1 [math.DG] 22 Jun 2011 JB -, JBW-, JBW -, JH -algebra, Ricatti equation, Riemann space, symmet- ric space, R-space, octonion plane, projective plane, Barbilian space, Tzitzeica equation, quantum group, B¨acklund-Darboux transformation, Hopf algebra, Yang-Baxter equation, KP equation, Sato Grassmann manifold, genetic alge- bra, random quadratic form. 1The author was partially supported from the contract PN-II-ID-PCE 1188 517/2009. 2 CONTENTS 1. Jordan structures ................................. ....................2 § 2. Algebraic varieties (or manifolds) defined by Jordan pairs ............11 § 3. Jordan structures in analysis ....................... ..................19 § 4. Jordan structures in differential geometry . ...............39 § 5. Jordan algebras in ring geometries . ................59 § 6. Jordan algebras in mathematical biology and mathematical statistics .66 § 7. -
Representation Theory
M392C NOTES: REPRESENTATION THEORY ARUN DEBRAY MAY 14, 2017 These notes were taken in UT Austin's M392C (Representation Theory) class in Spring 2017, taught by Sam Gunningham. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Kartik Chitturi, Adrian Clough, Tom Gannon, Nathan Guermond, Sam Gunningham, Jay Hathaway, and Surya Raghavendran for correcting a few errors. Contents 1. Lie groups and smooth actions: 1/18/172 2. Representation theory of compact groups: 1/20/174 3. Operations on representations: 1/23/176 4. Complete reducibility: 1/25/178 5. Some examples: 1/27/17 10 6. Matrix coefficients and characters: 1/30/17 12 7. The Peter-Weyl theorem: 2/1/17 13 8. Character tables: 2/3/17 15 9. The character theory of SU(2): 2/6/17 17 10. Representation theory of Lie groups: 2/8/17 19 11. Lie algebras: 2/10/17 20 12. The adjoint representations: 2/13/17 22 13. Representations of Lie algebras: 2/15/17 24 14. The representation theory of sl2(C): 2/17/17 25 15. Solvable and nilpotent Lie algebras: 2/20/17 27 16. Semisimple Lie algebras: 2/22/17 29 17. Invariant bilinear forms on Lie algebras: 2/24/17 31 18. Classical Lie groups and Lie algebras: 2/27/17 32 19. Roots and root spaces: 3/1/17 34 20. Properties of roots: 3/3/17 36 21. Root systems: 3/6/17 37 22. Dynkin diagrams: 3/8/17 39 23. -
Lecture 1: Group C*-Algebras and Actions of Finite Groups on C*-Algebras 11–29 July 2016
The Second Summer School on Operator Algebras and Noncommutative Geometry 2016 East China Normal University, Shanghai Lecture 1: Group C*-algebras and Actions of Finite Groups on C*-Algebras 11{29 July 2016 Lecture 1 (11 July 2016): Group C*-algebras and Actions of Finite N. Christopher Phillips Groups on C*-Algebras Lecture 2 (13 July 2016): Introduction to Crossed Products and More University of Oregon Examples of Actions. Lecture 3 (15 July 2016): Crossed Products by Finite Groups; the 11 July 2016 Rokhlin Property. Lecture 4 (18 July 2016): Crossed Products by Actions with the Rokhlin Property. Lecture 5 (19 July 2016): Crossed Products of Tracially AF Algebras by Actions with the Tracial Rokhlin Property. Lecture 6 (20 July 2016): Applications and Problems. N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 1 / 28 N. C. Phillips (U of Oregon) Group C*-Algebras, Actions of Finite Groups 11 July 2016 2 / 28 A rough outline of all six lectures General motivation The material to be described is part of the structure and classification The beginning: The C*-algebra of a group. theory for simple nuclear C*-algebras (the Elliott program). More Actions of finite groups on C*-algebras and examples. specifically, it is about proving that C*-algebras which appear in other Crossed products by actions of finite groups: elementary theory. parts of the theory (in these lectures, certain kinds of crossed product C*-algebras) satisfy the hypotheses of known classification theorems. More examples of actions. Crossed products by actions of finite groups: Some examples.